Subspace Clustering with a Twist

David Wipf, Yue Dong, Bo Xin Peking University
Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence, PMLR R14:485-494, 2016.

Abstract

Subspace segmentation or clustering can be defined as the process of assigning subspace labels to a set of data points assumed to lie on the union of multiple low-dimensional, linear subspaces. Given that each point can be efficiently expressed using a linear combination of other points from the same subspace, a variety of segmentation algorithms built upon $\ell_1$, nuclear norm, and other convex penalties have recently shown state-of-the-art robustness on multiple benchmarks. However, what if instead of observing the original data points, we instead only have access to transformed, or ‘twisted’ so to speak, measurements? Here we consider underdetermined affine transformations that may arise in computer vision applications such as bidirectional reflectance distribution function (BRDF) estimation. Unfortunately most existing approaches, convex or otherwise, do not address this highly useful generalization. To fill this void, we proceed by deriving a probabilistic model that simultaneously estimates the latent data points and subspace memberships using simple EM update rules. Moreover, in certain restricted settings this approach is guaranteed to produce the correct clustering. Finally a wide range of corroborating empirical evidence, including a BRDF estimation task, speaks to the practical efficacy of this algorithm.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR14-wipf16a, title = {Subspace Clustering with a Twist}, author = {Wipf, David and Dong, Yue and University, Bo Xin Peking}, booktitle = {Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence}, pages = {485--494}, year = {2016}, editor = {Ihler, Alexander and Janzing, Dominik}, volume = {R14}, series = {Proceedings of Machine Learning Research}, month = {25--29 Jun}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r14/main/assets/wipf16a/wipf16a.pdf}, url = {https://proceedings.mlr.press/r14/wipf16a.html}, abstract = {Subspace segmentation or clustering can be defined as the process of assigning subspace labels to a set of data points assumed to lie on the union of multiple low-dimensional, linear subspaces. Given that each point can be efficiently expressed using a linear combination of other points from the same subspace, a variety of segmentation algorithms built upon $\ell_1$, nuclear norm, and other convex penalties have recently shown state-of-the-art robustness on multiple benchmarks. However, what if instead of observing the original data points, we instead only have access to transformed, or ‘twisted’ so to speak, measurements? Here we consider underdetermined affine transformations that may arise in computer vision applications such as bidirectional reflectance distribution function (BRDF) estimation. Unfortunately most existing approaches, convex or otherwise, do not address this highly useful generalization. To fill this void, we proceed by deriving a probabilistic model that simultaneously estimates the latent data points and subspace memberships using simple EM update rules. Moreover, in certain restricted settings this approach is guaranteed to produce the correct clustering. Finally a wide range of corroborating empirical evidence, including a BRDF estimation task, speaks to the practical efficacy of this algorithm.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T Subspace Clustering with a Twist %A David Wipf %A Yue Dong %A Bo Xin Peking University %B Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2016 %E Alexander Ihler %E Dominik Janzing %F pmlr-vR14-wipf16a %I PMLR %P 485--494 %U https://proceedings.mlr.press/r14/wipf16a.html %V R14 %X Subspace segmentation or clustering can be defined as the process of assigning subspace labels to a set of data points assumed to lie on the union of multiple low-dimensional, linear subspaces. Given that each point can be efficiently expressed using a linear combination of other points from the same subspace, a variety of segmentation algorithms built upon $\ell_1$, nuclear norm, and other convex penalties have recently shown state-of-the-art robustness on multiple benchmarks. However, what if instead of observing the original data points, we instead only have access to transformed, or ‘twisted’ so to speak, measurements? Here we consider underdetermined affine transformations that may arise in computer vision applications such as bidirectional reflectance distribution function (BRDF) estimation. Unfortunately most existing approaches, convex or otherwise, do not address this highly useful generalization. To fill this void, we proceed by deriving a probabilistic model that simultaneously estimates the latent data points and subspace memberships using simple EM update rules. Moreover, in certain restricted settings this approach is guaranteed to produce the correct clustering. Finally a wide range of corroborating empirical evidence, including a BRDF estimation task, speaks to the practical efficacy of this algorithm. %Z Reissued by PMLR on 04 October 2026.
APA
Wipf, D., Dong, Y. & University, B.X.P.. (2016). Subspace Clustering with a Twist. Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R14:485-494 Available from https://proceedings.mlr.press/r14/wipf16a.html. Reissued by PMLR on 04 October 2026.

Related Material